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ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

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Numerical Investigation of the Optimal Measurement for a Semilinear Descriptor System with the Showalter-Sidorov Condition: Algorithm and Computational Experiment

Автор(ы):

Natalia Alexandrovna Manakova

South Ural State University,
Professor of the Department of Equations of Mathematical Physics,
Doctor of Physical and Mathematical Sciences

manakovana@susu.ru

Olga Gavrilova

South Ural State University,
Senior Lecturer of the Department of Equations of Mathematical Physics

gavrilovaov@susu.ru

Ksenia Vladimirovna Perevozchikova

South Ural State University,
Assistant of the Department of Equations of Mathematical Physics

vasiuchkovakv@susu.ru

Аннотация:

The article deals with the problem of optimal measurement for a semilinear descriptor system with a distinguished linear part and a nonlinear term unsolved with respect to the derivative of the unknown vector function with the Showalter–Sidorov initial condition. Basing on the methods of the theory of optimal control we found sufficient conditions for the existence of solutions of the optimal measurement problem – the problem of recovering a dynamically distorted signal from a measuring device. An algorithm for finding a numerical solution uses the methods of decomposition, penalty and the Ritz method as well. The algorithm is based on the representation of the measurement components by polynomials of a given degree, which allows reducing the optimal control problem to a computer programming problem with respect to the unknown coefficients of the polynomials. As an example of a sensor we consider Fitz Hugh – Nagumo oscilloscope described by a nonlinear descriptor system. Computational experiments for the considered sensor model are presented.

Ключевые слова

Ссылки:

  1. Granovsky, V. A. Dinamicheskie izmerenija. Osnovy metrologicheskogo obespechenija [Dynamic Measurements. Basics of Metrological Support]. Leningrad, Jenergoatomizdat, 1984
  2. Tihonov, A. N., Arsenin, V. Ja. Metody reshenija nekorrektnyh zadach[Methods for Solving Ill-Posed Problems]. Moscow, Nauka, 1979
  3. Shestakov, A. L. Modal Synthesis of a Measurement Transducer. Problemy upravlenija i informatiki, 1995, no. 4, pp. 67-75
  4. Derusso, P., Roj, R., Klouz, H. Prostranstvo sostojanij v teorii upravlenija [State Space in Control Theory]. Moscow, Nauka, 1970
  5. Shestakov, A. L., Sviridyuk, G. A. A New Approach to Measuring Dynamically Distorted. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2010, no. 16(192), pp. 116-120
  6. Sagadeeva, M. A., Zagrebina, S. A., Manakova, N. A. Optimal Control of Solutions of a Multipoint Initial-Final Problem for Non-Autonomous Evolutionary Sobolev Type Equation. Evolution Equations and Control Theory, 2019, vol. 8, no. 3, pp. 473-488
  7. Zamyshlyaeva, A. A., Manakova, N. A., Tsyplenkova, O. N. Optimal Control in Linear Sobolev Type Mathematical Models. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2020, vol. 13, no. 1, pp. 5-27. DOI: 10. 14529/mmp200101
  8. Zamyshlyaeva, A. A., Tsyplenkova, O. N. Optimal Control of Solutions of the Showalter-Sidorov-Dirichlet Problem for the Boussinesq-Love Equation. Diff erential Equations, 2013, vol. 49, no. 11, pp. 1356-1365
  9. Sviridyuk, G. A., Efremov, A. A. An Optimal Control Problem for a Class of Linear Equations of Sobolev Type. Russian Mathematics, 1996, vol. 40, no. 12, pp. 60-71
  10. Shestakov, A. L., Keller, A. V., Zamyshlyaeva, A. A., Manakova, N. A., Zagrebina, S. A., Sviridyuk,, G. A. The Optimal Measurements Theory as a New Paradigm in the Metrology. Journal of Computational and Engineering Mathematics, 2020, vol. 7, no. 1, pp. 3-23
  11. Keller, A. V. On the Computational Effi ciency of the Algorithm of the Numerical Solution of Optimal Control Problems for Models of Leontieff Type. Journal of Computational and Engineering Mathematics, 2015, vol. 2, no. 2, pp. 39-59
  12. Shestakov, A. L., Keller, A. V., Nazarova, E. I. Numerical Solution of the Optimal Measurement Problem. Automation and Remote Control, 2012, vol. 73, no. 1, pp. 97-104
  13. Sagadeeva, M. A., Bychkov, E. V., Tsyplenkova, O. N. The Pyt’ev-Chulichkov Method for Constructing a Measurement in the Shestakov-Sviridyuk Model. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2020, vol. 13, no. 4, pp. 68-82. DOI: 10. 14529/mmp200407
  14. Shestakov, A. L., Sviridyuk, G. A., Keller, A. V., Zamyshlyaeva, A. A., Khudyakov Y. V. Numerical Investigation of Optimal Dynamic Measurements. Acta IMEKO, 2018, vol. 7, no. 2, pp. 65-72
  15. Manakova, N. A. Method of Decomposition in the Optimal Control Problem for Semilinear Sobolev Type Models. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2015, vol. 8, no. 2, pp. 133-137. DOI: 10. 14529/mmp150212
  16. Fitz, Hugh R. Mathematical Models of Threshold Phenomena in the Nerve Membrane. Bulletin of Mathematical Biology, 1955, vol. 17, no. 4, pp. 257- 278
  17. Nagumo, J., Arimoto, S., Yoshizawa, S. An Active Pulse Transmission Line Simulating Nerve Axon. Proceedings of the IRE, 1962, vol. 50, no. 10, pp. 2061-2070
  18. Borisov, V. G. Parabolic Boundary Value Problems with a Small Parameter Multiplying the Derivatives with Respect to t. Mathematics of the USSR-Sbornik, 1988, vol. 59, no. 2, pp. 287-302
  19. Thompson, J. M. T. Instabilities and Catastrophes in Sciences and Engineering. Chichester, New York, Brisbane, Toronto, Singapore, John Wiley and Sons, 1982

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